Probability and Statistics Problems

Probability and Statistics Problems

Information for the Teacher

This section of problems has a common theme of Probability and Statistics. Probability is about likelihood and risk. Statistics is about making sense of data and uses probabilistic ideas such as randomness.

The “Starter” problems are for beginners and hints are provided. Most Year 8/9 or 10 students could cope with these problems. Students should work in small groups (2, 3 or 4) and be instructed to read the question, ask for any clarifications and then craft a solution to present to others. Once answered and explained the solution provided can be issued. Note that there are extensions on the solution page for each problem. These challenges will help build confident and lasting problem solving students.

How to Solve a Problem

1. Read and understand the problem, clarify the task.

2. Draw a picture of the situation.

3. Decide upon a strategy to proceed.

4. Solve using your strategy.

5. Present the solution and answer the task.

This method of progressing into a problem has been promoted for a long time (Polya 1945 How to Solve it!). I have been promoting this as a good habit to develop in all students and use in solving the tasks in all of the current mathematics and Statistics Achievement Standards.

A special note for the “5. Present the solution and answer the task.” Good communication is essential. Every problem presents an opportunity to practice the SRWL (Speak, Read, Write, Listen) of being literate. A good teacher tool is to randomly select a student from the group to do the presentation from this problem. Every student then has the expectation of performing and so is more engaged in the problem. Share different solutions!

Problem solving is developing a survival attitude to life. Developing a confidence of being able to overcome difficulty, work around an issue, think outside the square are all powerful attributes that build a more confident person. Struggling on a problem and developing perseverance may well be the most important aspect of this development. Learning from a mistake is vital and remember that a repeated mistake is stupidity. Mistakes are possibly the best situations of learning that we can experience.

“It is better to solve one problem five ways than five problems one way. “ Polya 1945.

Methods to Solve a Problem

There are many ways to solve a problem. In the appendix is a checklist that a student can use when a new method has been employed. The following is an incomplete list in not particular order of many solution methods. The list is incomplete because there is always another way!

1. Guess and Check.

Start with a hunch and see what happens. Make a better guess and continue. Monitor and feedback.

2. Scientific method

Reducing a problem to two variables, one you change and one you measure.

3. Drawing a picture

You can only draw the problem if you understand it. This helps solve it.

4. Listing all possibilities

This can be tedious but works.

5. A Counter Example

This will destroy any faulty conjectures.

6. Pattern Finding

The thinking needed is at NZC Level 4. Recursive patterns build on previous terms, nth term is multiplicative.

7. Working Backwards

An intuition is needed here. Mathematicians have excellent intuition by the way.

8. Reduce the problem in size

Making the problem smaller helps.

9. AHHA! Insights

This is insight at its best. The better a person is at problem solving the better they are at AHHA moments.

10. View from a different perspective

This is like thinking outside the square and allows a different view. Instead of seeing education as teaching, see it as learning.

11. Literature Review

Has this problem been done before?

12. Transpose the Problem

Move a number problem into geometry. Logarithms transpose multiplication to addition. Riemann transposed primes to complex numbers and this has since been made into a graph.

13. Ask Google!

Type your question into Google and see what happens. I wondered what value 0^0 had and discovered it is a major debate and could be 0 or 1.

14. Persevere

This is the best approach to any problem. No time is involved. Just keep going.

15. Share the problem

Verbalising a problem, discussing an idea, sharing thoughts is how mathematicians make progress. Very few of the ideas in mathematics resulted from one person in solitude.

The key thinking to develop in mathematics is strategic, logic, critical and creative. Strategic is “the method used”, logic is “the deduction process”, critical is “being reflective”, and creative is “where ideas generate and emerge”. Enjoyment and perseverance are important as well.

Share different solutions and reinforce multiple solution effort. Much is learned from solving a problem in different or better ways. Deep insight comes after deep thought.

Starter Problem #1 • Horse Racing

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The Finish Line

Instructions

The horses are numbered 1 to 13. Choose your horse number and a colour to keep track of the race.

Toss two dice and add up the result. The horse with that number moves ahead one square. Take turns to throw the dice.

A horse race is about winning so the first horse to cross the finish line wins. There is a 2nd, 3rd and 4th so keep racing as well.

MORE!

Have a few races and record what you notice.

Is there a winning strategy?

Can you explain your strategy for winning.

Craft your answer!

Be prepared to explain it to another person.

Solution Starter Problem #1 • Horse Racing

The curious thing with throwing 2 dice and adding the result is not all numbers are possible and the probability of the numbers that do show up are not the same. These are explainable patterns.

Two DICE

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6

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6

7

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12

The theoretical probability of a 3, for example, being tossed on any random throw is 2/36. There are two ways a 3 can be made and these are shown shaded in the table of possibilities above.

You might notice that 1 and 13 are not even possible so if you select those horses you will never win.

The most likely is horse number 7, followed by 6 and 8, followed by 5 and 9, followed by 4 and 10, followed by 3 and 11, and lastly with a only 1/36 chance are horses 2 and 12. Colouring in this pattern helps.

This is the most likely order the horses will finish and pretty much every time they will form an arrowhead with 7 in the lead every time, but not always. The nature of random does change things.

This is a fun activity and best later in the day when some more engaging activities are needed. Students can post winning strategies and have a great mathematical discourse about why. The real answers are as above and a thoughtful interpretation of the table is the target for each student.

Understand what is happening! None of this is magic but random effects do happen and on a very rare occasion, horse number 12 might well win. If it does, go and buy a LOTTO ticket immediately!

Problem #2 • Prison Cells

Read the rules and play!

Have a few games and see if you can create a strategy that will help you win more often.

Remember the difference between two numbers is the largest minus (or take away) the smallest. Eg the difference between 5 and 2 is 3, the difference between 2 and 5 is also 3.

Solution Problem #2 • Prison Cells

This is a curious game as well in much the same was as Horse Racing.

Two DICE

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6

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0

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0

From the table we see “1” is the most likely happening 10/36 times or nearly 30% . Next is “2” on 8/36. Next are “0” and “3” on 6/36, next “4” with 4/36 and lastly “5” on just 2/36.

This informs how you choose to place your counters.

A winning strategy might be to place 6 counters in Cell 1, 3 in Cell 2 and 1 in Cell 3. There may not be a perfect strategy to this game but simulating many games or perhaps collecting all the class results might give some insights.

Extensions

· add Cell 6

· change the calculation to addition

· use two 0-9 dice and rename cells 0 to 9.

· Add a double six rule to get everyone out all at once.

· have students invent a context for a similar game.

· Use the different shaped Dungeons and Dragons dice to make new games.

Start the discussion of “What is random?”

There is a lot of random in our lives and we do need a sensible interpretation of always and impossible. Listing on the wall all the random words used is a starter. Words like “might” and “can” and “beyond reasonable doubt”. Now assign a probability!

Problem #3 • Triangle of Dice

ACTION

Toss three dice and see if you can

make a triangle from the numbers.

EG, Toss 1,6,2 and if these were 1cm, 6cm, 2cm the answer is NO.

EG, Toss 6,6,6 and if these were 6cm, 6cm and 6cm the answer is YES.

TASK

What is the probability of throwing three dice and being able to form a triangle?

This problem is about tossing three dice and interpreting the numbers tossed as measures. The question is then asked about how likely it is to be able to form a triangle. How can you do this?

Craft your answer!

Be prepared to explain it to another person.

Solution Problem #3 • Triangle of Dice

This is a very cool problem. Listing all possibilities and deciding for each if a triangle can then be formed will solve it.

Firstly, what is the underlying knowledge here about when a triangle can be formed?

ANSWER = If two sides are longer than the other then a triangle is formed. If the two shorter sides are the same a line is formed, not a triangle. If the two shorter sides are less an open three sides shape is created.

The all possibilities answer shows 111/216 = 51.4% or just over half of the time. The chart is too long to list here as it contains 6 x6x6 = 216 items. 111 combinations form a triangle.

A simulation shows after 1000 throws a confirming answer. This is an excellent exercise to give students.

BY using PPDAC and tossing a sample of about 30 to 50 will also give an answer but notice from the simulation that samples less than 400 are a bit variable and will only give an estimate.

Problem #4 • The 90 Product

ACTION

Task

What is the probability of tossing three dice and getting a product of 90?

Toss three dice and multiply the resulting numbers.

This is a curious problem because only some numbers can be the product and 90 is one of them. Even more curious is that some ways of making 90 are not possible with 6 sided dice.

Investigate and see what you find out.

EXTENSION

This problem was found on https://brilliant.org/daily-problems/ and it is free to sign up and do some of the challenges.

The original problem is stated a little bit differently and asks a slightly different question.

Three fair cubical dice are thrown. If the probability that the product of the scores on the three dice is 90 is a/b , where a and b are positive coprime integers, then find the value of (b – a) .

Co-prime means two numbers only have 1 as a common factor. The numbers 2 and 4 are not co-prime, 1 and 2 are.

Solution Problem #4 • The 90 Product

A table can be made showing all the possible 6x6x6 = 216 products. It would need to be in 3d and this can be done on Xcel.

Another way is to investigate the factors of 90.

F90 = {1,2,3,5,6,9,10,15,18,30,45,90}

The interesting combinations are 3x30, 5x18 and 6x15. These all factorise to 3x5x6.

There are 3 ways of the first dice having one of these numbers.

There are 2 ways for the remaining dice to have another factor.

There is 1 way left for the remaning dice to have the correct number.

This is an application of 3!=3x2x1 or the multiplication principle.

This means there are 6 ways from 216 that will give 90. 6/216 simplifies to 1/36 as a co-prime simplified fraction.

The answer to the extension then, is, 35 or 36-1.

All of this is explained on https://brilliant.org/problems/three-dice-thrown-simultaneously/ .

Problem #5 • Sample Size

This problem is very important and everyone

needs to spend some time getting their heads around it.

Understanding how big a sample should be before making grand summary statements and judgements will be very useful.

TASK

How big should a sample be to use to get a reliable answer?

ACTION (here is some help)

Toss a dice 5 times and calculate the probability of throwing a head.

Now make up a table and see what happens when the number of tosses changes from 2 to 40 or so.

Number of Tosses

Number of heads

Prob(Head)

2

3

4

5

10

20

40

You could investigate what happens for the missing numbers.

Now draw a graph of your results. Tosses on the abscissa (x-axis) and Prob(Heads) in the ordinate (y-axis).

What do you notice? Discuss with a buddy and try and write down what you are thinking.

Answer the TASK question…

This task came about because of small sample sizes in research articles and the big claims made about those small groups. Small sample sizes are problematic. There is simply too much variation in the outcome to make reliable judgmental statements.

The Margin of Error is directly related to the inverse square of the sample size.

MOE = 1/(√n) where n is the sample size.

For Example

The MOE for a sample of size 4 is 1/√4 = ½ = 50%

There is a 50% error just as a result of the sample size.

The MOE for a sample of size 1000 = 1/√(1000)=3.16%

This is the error on polls you see in the newspaper after ColmarBrunton has surveyed 1000 people.

Here is a computer simulation of 2018 tosses of a coin. The probability waves around until about 200 tosses and then slowly, but surely, creeps its way back to 50%. The small sample size variation is clearly visible.

Here a measure of variation in samples of different sizes is plotted against sample size. The data is the Lake Taupo Trout 1993 available on Census@Schools. The software used to explore this was iNZight and Excel.

Clearly the sample size before variation starts to settle is once again more like 50 or so at least. It is probably a waste of time to explore 200+ samples but with computer technology now this is actually not a time consuming task so just use large samples.

The data collected by VISA and GOOGLE is in the millions so pretty much anything they notice is probably true!

Problem #6 • Putting Probability Distribution

If you have played golf you will appreciate that variation happens. There are over 200 muscles in your body that need to coordinate. Add the complication of feeling and mood and then include the trickery of the contour of the green and it is no wonder we all have so much trouble being perfect on every shot.

Question

How far are you from the hole on a green when your probability of getting the putt is about 50%.

ACTION (here is some help)

Find a few golf balls, a putter a nice flat green, a few tees and a tape measure. Mark of 10cm, 20cm, 30cm, 40cm etc from the hole with the tees and then putt 10 or so balls from each position. Build up a table of %success of getting the ball in the hole. You may need to graph the results to estimate the answer to the question above. The following table might help.

Distance

Success /total shots

%success

10cm

20cm

30cm

Extension

On the practice range do a similar experiment with chipping. It is a very useful exercise to know pretty much how far each chipping club will hit a ball. The chip can be with a different amount of swing (1/8 back, ¼ back etc, accelerate and smooth follow through).

This task is a combination of statistics and probability. I hope you play golf!

Problem #6 • Putting Probability Distribution

My results

My estimate is when I am about one metre from the hole I have a 50% chance of getting the putt. This seems to be right when I play. Some days better somedays worse and that seems to depend on how much I have on my mind! Concentration!

My sand wedge, wedge, 9 and 8 iron all combined with 1/8, ¼, 2/3, half swing and full swing give me a reasonably reliable outcome with range from about 2m to 100m and everything in between. Getting the choice of club right is as important as making sure the stance, the target and relaxed swing are all present as well. Being quite optimistic about the shot is also present.

Golf is a great personal challenge. I play on a 16 handicap meaning I need just under 1 extra shot per hole to complete the course in regulation or 72 shots. The variation on that score is about 3 shots over or under about 80% of the time. When I play a game I am 80% certain of scoring between 69 and 75 after the handicap has been removed. My gross score is almost always between 85 and 91 or in 9 hole game between 41 and 47.

Every Golf Club needs young members and I would encourage all school students to take advantage of the free lessons and membership offered across the country. Golf is a great social activity and can be taken as seriously as you choose it be. My golf is pretty relaxed and I like the company, the personal challenge and the exercise. If I have a low scoring game that is a bonus!

Good problems that show a different way of approaching a problem are collectible items. Please send any you have discovered to jimhogan2@icloud.com

Other sources of problems are

Australian Mathematics Competition problems

http://www.amt.edu.au/wuamc.html

American Problem Solving

http://wiki.artofproblemsolving.com/index.php/Problem_solving

http://wiki.artofproblemsolving.com/index.php/AMC_Problems_and_Solutions

Competitions for Mathematics

http://en.wikipedia.org/wiki/List_of_mathematics_competitions

Some good books of problems include

Just have a look on Amazon!

Check out Math Mind Questions from previous years on www.bopma.org.nz

HG

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Published Feb 2021